4.5.5 Remainders with a^p, b^p, and ab^p

These problems use the principle that “the remainders after algebra equals the algebra of the remainders.”

Example

Problem: If $6x^7$ has remainder 2 and $2y^7$ has remainder 3, what is the remainder of $4xy^7$?

Solution:

Multiply the expressions: $6x^7 \times 2y^7 = 12xy^7$

Divide by 3: $12xy^7 \div 3 = 4xy^7$

Same operations with remainders: $2 \times 3 = 6 \div 3 = 2$

Answer: 2

When You Get a Fractional Remainder

If algebra on remainders gives a fraction, find small values of $x$ and $y$ that work, then compute directly.

Example: If $2x^7$ has remainder 1 and $y^7$ has remainder 3, find remainder of $xy^7$.

Multiplying and dividing: $xy^7 \cong 1.5$ (fractional!)

Find $x = 4$ (from first expression) and $y = 3$ (from second)

So $xy^7 = 12^7$ which has remainder 5

---

Problem Set 4.5.5

Practice these problems. Type your answer and press Enter to check: